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Mirrors > Home > MPE Home > Th. List > tnggrpr | Structured version Visualization version GIF version |
Description: If a structure equipped with a norm is a normed group, the structure itself must be a group. (Contributed by AV, 15-Oct-2021.) |
Ref | Expression |
---|---|
tngngp3.t | ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) |
Ref | Expression |
---|---|
tnggrpr | ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑇 ∈ NrmGrp) → 𝐺 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tngngp3.t | . . . . 5 ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) | |
2 | eqid 2728 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
3 | 1, 2 | tngbas 24545 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → (Base‘𝐺) = (Base‘𝑇)) |
4 | eqidd 2729 | . . . 4 ⊢ (𝑁 ∈ 𝑉 → (Base‘𝐺) = (Base‘𝐺)) | |
5 | eqid 2728 | . . . . . . 7 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
6 | 1, 5 | tngplusg 24547 | . . . . . 6 ⊢ (𝑁 ∈ 𝑉 → (+g‘𝐺) = (+g‘𝑇)) |
7 | 6 | eqcomd 2734 | . . . . 5 ⊢ (𝑁 ∈ 𝑉 → (+g‘𝑇) = (+g‘𝐺)) |
8 | 7 | oveqdr 7443 | . . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g‘𝑇)𝑦) = (𝑥(+g‘𝐺)𝑦)) |
9 | 3, 4, 8 | grppropd 18902 | . . 3 ⊢ (𝑁 ∈ 𝑉 → (𝑇 ∈ Grp ↔ 𝐺 ∈ Grp)) |
10 | 9 | biimpd 228 | . 2 ⊢ (𝑁 ∈ 𝑉 → (𝑇 ∈ Grp → 𝐺 ∈ Grp)) |
11 | ngpgrp 24502 | . 2 ⊢ (𝑇 ∈ NrmGrp → 𝑇 ∈ Grp) | |
12 | 10, 11 | impel 505 | 1 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑇 ∈ NrmGrp) → 𝐺 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ‘cfv 6543 (class class class)co 7415 Basecbs 17174 +gcplusg 17227 Grpcgrp 18884 NrmGrpcngp 24480 toNrmGrp ctng 24481 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-er 8719 df-en 8959 df-dom 8960 df-sdom 8961 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-plusg 17240 df-tset 17246 df-ds 17249 df-0g 17417 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-grp 18887 df-ngp 24486 df-tng 24487 |
This theorem is referenced by: tngngp3 24567 |
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